For Lingering Souls once the spell is played on my turn it is put into the graveyard. However if I have enough mana can I instantly play the flashback ability or do I need to wait until my next turn to take it out the graveyard?
If the card is in the graveyard, you have the mana to pay the flashback cost, and priority to play the spell, you can play it. Note that you have to wait for the spell to resolve before you can play it again, because that's when you put the card in the graveyard.
There is no rule specifying that you have to wait until your next turn.